Final Report Summary - MUSYX (Multiscale Simulation of Crystal Defects)
The overarching purpose of this project is to develop a comprehensive numerical analysis theory for multi-scale coupling methods to model the atomistic structure of material defects. The project develops numerical algorithms, rigorous numerical analysis, and prototype software, for the efficient numerical simulation of material defects. The project is split into four related themes, with the following main results achieved:
Theme A: Stability and bifurcation. The problem to be addressed in this theme
are sharp stability estimates for atomistic/continuum (A/C) coupling methods and the consequence of such results for the a priori error analysis of A/C methods. The first major result was to characterise for which A/C coupling methods stability of the underlying atomistic model implies stability of the A/C scheme. The second main result of this Theme is the development of a rigorous bifurcation theory for crystalline defects, specifically in the context of fracture.
Theme B: Transition rates. Theme B concerns the approximation error analysis of saddle points (as opposed to minima in the bulk of the A/C coupling literature) and of transition rates in the harmonic approximation, which are expressed in terms of energy and vibrational entropy differences between energy minima and saddle points. The first main result in this theme is a sharp rigorous convergence theory for equilibria that are not minima. In addition it turned out that these convergence results were required not only in energy norms but also max-norms, which required the development of new discrete regularity tools. Secondly, a sharp convergence analysis of vibrational entropy differences applicable to both formation energies and to transition rates for developed.
Theme C: Temperature. The goal of this theme is to develop models and approximation results for defect formation free energy. In a a first paper a one-dimensional toy model and an extensive analysis have been developed that establishes the first steps towards an approximation error analysis for free energy with explicit convergence rates. In the two and three-dimensional setting analogous results have been established for in the harmonic approximation setting. These results and the technical tools established with them provide the tool for a new approach to finite-temperature coarse-graining.
Theme D: MM/MM and QM/MM Coupling. The aim of Theme D is to explore QM/MM (quantum to classical) coupling methods. We have discovered a new decomposition of total potential energy for electronic structure models into spatially localised contributions. We then used this result to develop a new class of QM/MM coupling schemes with guaranteed and controllable approximation errors. Together these results kick off an extremely promising new direction for the investigation of interatomic potentials and QM/MM multi-scale models and related problems.
Theme A: Stability and bifurcation. The problem to be addressed in this theme
are sharp stability estimates for atomistic/continuum (A/C) coupling methods and the consequence of such results for the a priori error analysis of A/C methods. The first major result was to characterise for which A/C coupling methods stability of the underlying atomistic model implies stability of the A/C scheme. The second main result of this Theme is the development of a rigorous bifurcation theory for crystalline defects, specifically in the context of fracture.
Theme B: Transition rates. Theme B concerns the approximation error analysis of saddle points (as opposed to minima in the bulk of the A/C coupling literature) and of transition rates in the harmonic approximation, which are expressed in terms of energy and vibrational entropy differences between energy minima and saddle points. The first main result in this theme is a sharp rigorous convergence theory for equilibria that are not minima. In addition it turned out that these convergence results were required not only in energy norms but also max-norms, which required the development of new discrete regularity tools. Secondly, a sharp convergence analysis of vibrational entropy differences applicable to both formation energies and to transition rates for developed.
Theme C: Temperature. The goal of this theme is to develop models and approximation results for defect formation free energy. In a a first paper a one-dimensional toy model and an extensive analysis have been developed that establishes the first steps towards an approximation error analysis for free energy with explicit convergence rates. In the two and three-dimensional setting analogous results have been established for in the harmonic approximation setting. These results and the technical tools established with them provide the tool for a new approach to finite-temperature coarse-graining.
Theme D: MM/MM and QM/MM Coupling. The aim of Theme D is to explore QM/MM (quantum to classical) coupling methods. We have discovered a new decomposition of total potential energy for electronic structure models into spatially localised contributions. We then used this result to develop a new class of QM/MM coupling schemes with guaranteed and controllable approximation errors. Together these results kick off an extremely promising new direction for the investigation of interatomic potentials and QM/MM multi-scale models and related problems.